[Aztlan] Plaster and Echoes

[David Lubman]

Greg asked how the "chirp" is produced.

I know of three models to explain chirp production. Chirps are produced by 
the interaction of sound with the staircase. Of these, the picket fence 
model provides the easiest physical insight into chirp production.

The response of the staircase to excitation with a short burst of sound is a 
series of 91 brief impulses, one for each stair.  A perceptual psychologist 
might explain the downward chirp by invoking the "picket fence" effect.

An interesting feature of auditory perception is that periodic sequences of 
impulsive sounds are perceived as tone bursts. The tonal period is the time 
between impulses. The tonal frequency is 1 divided by that period. For 
example, a sequence of impulses spaced 1 millisecond (0.001 sec) apart 
sounds like a 1000 Hz tone burst. A tonal pitch can be detected with as few 
as 3-5 impulses.

To this point we have explained the tone but not its downward glide (chirp). 
To explain the chirp, consider that sound is returned to the observer first 
from the lower steps because they are closer to the observer, and later from 
the higher steps because they are farther from the observer.

For the lower steps, sound "rays" emanating from handclaps are approximately 
parallel to staircase treads. The time interval between impulses for the 
lower steps is the time for a round trip traverse of a tread. Since the 
tread length T is about 0.262 m and the speed of sound c is about 343 m/sec 
the time between impulses is about 2T/c and the picket fence frequency is 
about c/2T = 343/0.524 = 654 Hz. This is the chirp starting frequency.

For the higher steps the time interval between impulses is greater and thus 
the frequency is lower. How low depends on the distance of the observer to 
the staircase. For the limiting case of an observer standing next to the 
staircase, sound rays are parallel to the hypotenuse of a right triangle 
whose two sides are the tread length (26.2 cm) and the riser height (26.4 
cm). The hypotenuse H is calculated by the Pythagorean theorem as 37.2 cm. 
The time between impulses is twice the time required for a round trip 
traverse of the hypotenuse ,or 2H/c. The limiting frequency is c/2H = 
343/0.744 = 461 Hz.

Thus the picket fence tone starts at about 654 Hz and glides down to as low 
as 461 Hz for observers close to the staircase. The starting frequency for 
observers at any distance from the staircase is the same (654 Hz). But the 
ending frequency increases with distance from the staircase to somewhere 
between 461 Hz and 654 Hz.

This simplification does not explain the harmonics which are also present in 
both the chirped echo and actual quetzal chirps (sonograms at 
http://www.ocasa.org/MayanPyramid2.htm). Harmonics enrichen the timbre of 
the chirp. Harmonics up to the 4th order are visible in these sonograms. The 
second and third harmonics are the strongest components.

The chirped echo and all of its harmonics are fully accounted for in two 
other math models. (1) Diffraction, in which the staircase is modeled as a 
reflective diffraction grating. Each stair is a line in that grating. (2) 
The scattering-convolution (S-C) model which is much simpler 
computationally. The S-C model can calculate the staircase response to ANY 
sound. Not just handclaps. It can also instantly auralize the echo. This 
feature is useful to ethnomusicologists and others who want to hear the echo 
response to voices, or to ethnic sound instruments such as rattles or turtle 
drums.

Best regards,

David Lubman
------------------

----- Original Message ----- 
From: "Greg Sandor" <gregory_sandor at hotmail.com>
To: "David Lubman" <dlubman at ix.netcom.com>; "D. Clark Wernecke" 
<CWernecke at compuserve.com>
Cc: "Aztlan" <aztlan at lists.famsi.org>; "D. Clark Wernecke" 
<102402.2332 at compuserve.com>
Sent: Friday, September 05, 2008 9:36 PM
Subject: Re: [Aztlan] Plaster and Echoes


> Can someone please describe how the "chirp" is produced?
>
> Regards,
>
> Greg
>
> (614) 517-7204
> greg at gregsandor.com
> http://www.gregsandor.com